IEV: Calculating Expected Offspring | Ben Cunningham

# IEV: Calculating Expected Offspring

Problem by Rosalind · on December 4, 2012

For a random variable $X$ taking integer values between 1 and $n$, the expected value of $X$ is $\mathrm{E}(X) = \sum_{k=1}^{n}{k \times \mathrm{Pr}(X = k)}$. The expected value offers us a way of taking the long-term average of a random variable over a large number of trials.

As a motivating example, let $X$ be the number on a six-sided die. Over a large number of rolls, we should expect to obtain an average of 3.5 on the die (even though it’s not possible to roll a 3.5). The formula for expected value confirms that $\mathrm{E}(X) = \sum_{k=1}^{6} k \times \mathrm{Pr}(X = k) = 3.5$.

More generally, a random variable for which every one of a number of equally spaced outcomes has the same probability is called a uniform random variable (in the die example, this “equal spacing” is equal to 1). We can generalize our die example to find that if $X$ is a uniform random variable with minimum possible value $a$ and maximum possible value $b$, then $\mathrm{E}(X) = \frac{a+b}{2}$. You may also wish to verify that for the dice example, if $Y$ is the random variable associated with the outcome of a second die roll, then $\mathrm{E}(X+Y) = 7$.

Given: Six nonnegative integers, each of which does not exceed 20,000. The integers correspond to the number of couples in a population possessing each genotype pairing for a given factor. In order, the six given integers represent the number of couples having the following genotypes:

1. AA-AA
2. AA-Aa
3. AA-aa
4. Aa-Aa
5. Aa-aa
6. aa-aa

Return: The expected number of offspring displaying the dominant phenotype in the next generation, under the assumption that every couple has exactly two offspring.

## Sample Dataset

1 0 0 1 0 1


## Sample Output

3.5


# R

library(dplyr)

f <- "iev.txt"

pop <- data_frame(
p = c(1, 1, 1, 0.75, 0.5, 0),
n =
sum(2 * pop$n * pop$p) %>%

3.5